To probe the role of graph architecture in the energy required for optimal control trajectories, we calculated the determinant ratio, which is defined as the ratio of the strength to the homogeneity of the connections between the first degree driver (anything with a non-zero entry in \(\mathbf {B}\) ) and the non-driver (anything with a zero entry in \(\mathbf {B}\) ) [1]}. This metric was derived assuming that a system has a greater number of driver nodes than non-driver nodes, and that the initial and final states are distributed around zero; importantly, both assumptions are accurate for our simulations. Quantitatively, the trade-off between strength and homogeneity is embodied in the ratio between the determinant of the Gram matrix of all driver to non-driver connections, and the determinant of that same matrix with each non-driver node removed iteratively. The gram matrix here is the inner product of the vectors giving connections from driver nodes to and non-driver nodes. More specifically, if \(C\)  is the Gram matrix of all driver to non-driver connections, and \(C_{k}\)  is the matrix of all connections from driver nodes to all but the \(k^{\mathrm {th}}\)  non-driver node, the determinant ratio is defined by \(N^{-1}\sum _{k=1}^{N}\frac{det(C_{k})}{det(C)}\) . Since the calculation of the determinant of large matrices can be computationally challenging, we use the equivalent estimate of the trace of the inverse of the Gram matrix, \(\mathrm {Trace}(C^{-1})\) , to calculate the average determinant ratio (see Kim et. al. for a full derivation) [1]}.
